Frozen-Orbital Approximation

A Hbasis functions provides K molecular orbitals, but lUJiW of these will not be occupied by smy electrons they are the virtual spin orbitals. If u c were to add an electron to one of these virtual orbitals then this should provide a means of calculating the electron affinity of the system. Electron affinities predicted by Konpman s theorem are always positive when Hartree-Fock calculations are used, because fhe irtucil orbitals always have a positive energy. However, it is observed experimentally that many neutral molecules will accept an electron to form a stable anion and so have negative electron affinities. This can be understood if one realises that electron correlation uDiild be expected to add to the error due to the frozen orbital approximation, rather ihan to counteract it as for ionisation potentials. 

Fukui functions and other response properties can also be derived from the one-electron Kohn-Sham orbitals of the unperturbed system [14]. Following Equation 12.9, Fukui functions can be connected and estimated within the molecular orbital picture as well. Under frozen orbital approximation (FOA of Fukui) and neglecting the second-order variations in the electron density, the Fukui function can be approximated as follows [15] ... 

Parr immediately pointed out that, in the frozen orbital approximation, these derivatives can be approximated with the squares of the lowest unoccupied (LUMO) and highest occupied molecular orbitals (HOMO) ... 

All these functional derivatives are well defined and do not involve any actual derivative relative to the electron number. It is remarkable that the derivatives of the Kohn-Sham chemical potential /rs gives the so-called radical Fukui function [8] either in a frozen orbital approximation or by including the relaxation of the KS band structure. On the other hand, the derivative of the Kohn-Sham HOMO-FUMO gap (defined here as a positive quantity) is the so-called nonlinear Fukui function fir) [26,32,50] also called Fukui difference [51]. 

Table I Interelectronic repulsion energy in the frozen orbital approximation (C) and in the SCF approximation (C )(l).

Photoelectron spectral measurements have prompted high-accuracy near-Hartree-Fock calculations on the Is hole states of 02. 261 Calculations were reported at Re for molecular O2. The frozen-orbital approximation evaluated the energy of Oj from the RHF calculations of Schaefer250 reported above. Then the IP are the difference between the O2 ground-state energy and the Ot energy. The IP obtained was 563.5 eV. Direct hole-state calculations for the relevant states of OJ, with the MO constrained to be of g or a symmetry, were also carried out. For the orbital occupancy (16), the computed IP was 554.4 eV. Finally, the restriction to g and u... 

We see from Eq. (3) that in addition to its use of approximate orbitals, Fukui s frontier density is a frozen-orbital approximation to the Fukui function, as indicated by the second term on the right-hand side of Eq. (3) [32]. 

The A -shell x-ray emission rates of molecules have been calculated with the DV-Xa method. The x-ray transition probabilites are evaluated in the dipole approximation by the DV-integration method using molecular wave functions. The validity of the DV-integration method is tested. The calculated values in the relaxed-orbital approximation are compared with those of the frozen-orbital approximation and the transition-state method. The contributions from the interatomic transitions are estimated. The chemical effect on the KP/Ka ratios for 3d elements is calculated and compared with the experimental data. The excitation mode dependence on the Kp/Ka ratios for 3d elements is discussed. 

These simple examples clearly show that orbital relaxation is crucial in vibronic coupling. Therefore, variationally optimized wavefunctions should be employed for vibronic coupling calculations. The frozen orbital approximation is not suitable for calculation [35]. 

The measured E value is directly proportional to the difference Eb(IE) = Ef — E,. The final state in PES consists of an ion and the outgoing photoelectron. The electronic structure of material is often described by approximate, one-electron wavefunctions (MO theory). MO approximation neglects electron correlation in both the initial and final states, but fortunately this often leads to a cancellation of errors when Ef, is calculated. A related problem arises when one tries to use the same wavefunctions to describe 4q and I f. This frozen orbital approximation is embedded in the Koopmans approximation (or the Koopmans theorem as it is most inappropriately called), equation 4,... 

Excitation energies calculated with the RPA and TDA approaches for N2 with a moderately large basis set are listed in Table 23. Both the RPA and TDA excitation energies are significantly lower than those obtained with the simplest frozen orbital approximation. All these approaches differ only in their treatment of the final state, and the pattern of predicted excitation energies shows this in a rather dramatic way. Both the RPA and TDA allow for a limited amount of relaxation and provide much improved predictions. Inclusion of the... 

As roughly sketched in the introduction, metabolic hydroxylation mediated by cytochrome P450 is attributed to a ferro-oxyl species, called Compoimd I (Cpd I). From present knowledge, Cpd I may be seen as an electrophilic oxidant [6], Thus fhe/ function (calculated for the substrates) should help identify those positions in a molecule, which are susceptible to an attack by Cpd I. Conversely, the function, evaluated for Cpd I, should show where Cpd I prefers to be attacked by the substrate. Atomic HOMO coefficients from semiempirical calculations on agrochemicals have already been quite successfully correlated to oxidative metabolic pathways [33-34], This procedure is essentially equivalent to calculating the/ function in the frozen orbital approximation, in whichreduces to the HOMO density. In the examples section, a nice demonstration of the breakdown of this approximation will be given. 

This example shows that the frozen orbital approximation, which is crucial for classical frontier orbital theory, can lead to severe artifacts and misleading interpretations. Fukui functions, on the other hand, capture electronic relaxation resulting from changes in the number of electrons. 

The composition of this review is as follows Section 2 describes the numerical examples of the rules for degenerate excitations. The data in the next section are obtained by highly correlated methods, since the effects of electron correlations are essential for accurate descriptions of the excited states. Section 3 demonstrates the interpretation of the rules by using the simplified model that corresponds to the frozen-orbital approximation (FZOA) [4]. In the excitation energy formulas to which the FZOA leads, the splitting schemes are related to the specific two-electron integrals, whose values are qualitatively analyzed by the relevant orbital characters. Finally, the summary is addressed in Section 4. 

The frozen-orbital approximation is explained in the standard texts of quanmm chemistry, such as A. Szabo, N. S. Ostlund, Modem Quantum Cchemistry Introduction to Advanced Electronic Structure Theory (McGraw-Hill, New York, 1989) the concept of the frozen-orbital analysis was first proposed in the following papen H. Nakai, H. Morita, H. Nakatsuji, J. Phys. Chem. 100, 15753 (1996)... 

We close with some additional words on the significance of the Fukui function. The Fukui function shores up the theoretical foundations of frontier molecular orbital theory. Equations (43)-(46) reveal that the site reactivity indices of frontier molecular orbital theory (Eqs. 47-49) may be regarded as the frozen orbital approximation to the Fukui function. The Fukui function is the zeroth-order index for site reactivity various functional derivatives of the Fukui function represent higher-order corrections to the zeroth-order site reactivity map provided by the Fukui function. For instance, the first-order correction to the Fukui function is... 

Let s assume that the MO model is a valid first-order approximation and that the radiation field causes a single electron to be promoted from the occupied MO , to the MO ground electronic state. Furthermore, the MOs in the ground and excited states are the same (the frozen orbital approximation) and there is no interaction between excited states so that an excited state can be represented by one excitation, j . This is the so-called singletransition approximation (STA). " ... 

The acronyms representing the algebraic expressions will be explained below. The first term (ORX), which is always negative, arises from the presence of single excitations a r in the second-order energy of the N — l)-particle system. Because of Brillouin s theorem, such excitations would not have contributed if we had used the HF orbitals of the (N — l)-particle system. Their effect is to optimize or relax the orbitals we did use (i.e., the HF orbitals of the iV-particle system) and hence this term is said to arise from orbital relaxation (ORX). Since it lowers the energy of the (N — l)-particle system relative to the frozen orbital approximation, it decreases the Koopmans theorem IP. Consider the second term (PRX), which is also always negative. 

The polarization effects compensating partly for exchange repulsion in the frozen orbital approximation discussed earlier are also found in the extended Hiickel method. 

---